66 Chapter 2 Systems of Linear Equations

In Problems 15–42, solve each system of equations. If the system has no solution, say that it is inconsistent.

15. 16. 17. 18.

19. 20. 21. 22.

23. 24 25. 26.

27. 28. 29. 30.

31. 32. 33. 34.

35. 36. 37. 38.

39. 40. 41. 42.

Applications

c x + 4y - 3z = -8

3x - y + 3z = 12

x + y + 6z = 1

c x + 2y - z = -3

2x - 4y + z = -7

-2x + 2y - 3z = 4

c x - y + z = -4

2x - 3y + 4z = -15

5x + y - 2z = 12

c x + y - z = 6

3x - 2y + z = -5

x + 3y - 2z = 14

c 3x - 2y + 2z = 6

7x - 3y + 2z = -1

2x - 3y + 4z = 0

c 2x - 2y + 3z = 6

4x - 3y + 2z = 0

-2x + 3y - 7z = 1

c 2x - 3y - z = 0

3x + 2y + 2z = 2

x + 5y + 3z = 2

c x - y - z = 1

-x + 2y - 3z = -4

3x - 2y - 7z = 0

c 2x - 3y - z = 0

-x + 2y + z = 5

3x - 4y - z = 1

c x - y - z = 1

2x + 3y + z = 2

3x + 2y = 0

c 2x + y - 3z = -2

-2x + 2y + z = -9

3x - 4y - 3z = 15

c x - 2y + 3z = 7

2x + y + z = 4

-3x + 2y - 2z = -10

c 2x + y = -4

-2y + 4z = 0

3x - 2z = -11

c x - y = 6

2x - 3z = 16

2y + z = 4

d 1

3 x -

3

2 y = -5

3

4 x +

1

3 y = 11

d 1

2 x +

1

3 y = 3

1

4 x -

2

3 y = -1

c 1

2 x + y = -2

x - 2y = 8

c 2x + 3y = 6

x - y =

1

2

b 3x - 2y = 0

5x + 10y = 4b 2x - 3y = -1

10x + y = 11

b 3x - y = 7

9x - 3y = 21b x + 2y = 4

2x + 4y = 8b x - y = 5

-3x + 3y = 2b 2x + y = 1

4x + 2y = 3

c 2x + 4y =

2

3

3x - 5y = -10

b 3x - 6y = 2

5x + 4y = 1b x + 3y = 5

2x - 3y = -8b 5x - y = 13

2x + 3y = 12

43. Dimensions of a Floor The perimeter of a rectangular floor is90 feet. Find the dimensions of the floor if the length is twicethe width.

44. Dimensions of a Field The length of fence required to enclosea rectangular field is 3000 meters. What are the dimensions

11.

x = 1, y = -1, z = 2

c 3x + 3y + 2z = 4

x - y - z = 0

2y - 3z = -8

12.

x = 2, y = -3, z = 1

c 4x - z = 7

8x + 5y - z = 0

-x - y + 5z = 6

13.

x = 2, y = -2, z = 2

c 3x + 3y + 2z = 4

x - 3y + z = 10

5x - 2y - 3z = 8

14.

x = 4, y = -3, z = 2

c 4x - 5z = 6

5y - z = -17

-x - 6y + 5z = 24

of the field if the difference between its length and width is 50 meters?

45. Agriculture According to the Illinois Farm Business Manage-ment Association, in 2008 in the state of Illinois, the nonlandproduction cost for planting corn was $512 per acre and the

Skill Building

In Problems 7–14, decide whether the values of the variables listed are solutions of the system of equations.

7.

x = 2, y = -1

b 2x - y = 5

5x + 2y = 88.

x = 2, y = 4

b 3x + 2y = 2

x - 7y = -309.

x = 2, y =

1

2

c 3x + 4y = 4

1

2 x - 3y = -

1

2

10.

x = -

1

2, y = 2

d 2x +

1

2 y = 0

3x - 4y = -

19

2

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2.1 Systems of Linear Equations: Substitution; Elimination 67

nonland cost for planting soybeans was $336 per acre.The average Illinois farm used 368 acres of land to raise cornand soybeans and budgeted $146,671 for planting these crops.If all the land and all the money budgeted is used, how manyacres of each crop should be planted?

Source: Illinois Farm Business Management Association

46. Movie Theater Tickets The Coral movie theater charges $9.00for adults and $7.00 for senior citizens. On a day when 325people paid an admission, the total receipts were $2495. Howmany who paid were adults? How many were seniors?

47. Mixing Nuts A store sells cashews for $5.00 per pound andpeanuts for $1.50 per pound. The manager decides to mix 30pounds of peanuts with some cashews and sell the mixture for$3.00 per pound. How many pounds of cashews should bemixed with the peanuts so that the mixture will produce thesame revenue as would selling the nuts separately?

48. Financial Planning A recently retired couple needs $6000 peryear to supplement their Social Security. They have $150,000to invest to obtain this income. They have decided on two investment options: Corporate bonds yielding 6% per annumand a Bank Certificate yielding 3% per annum.

(a) How much should be invested in each to realize exactly$6000?

(b) If, after two years, the couple requires $7000 per year inincome, how should they reallocate their investment toachieve the new amount?

49. Cost of Food in Japan In Osaka, Japan, the cost of three 1-liter cartons of whole milk and two 330-gram packages oftofu is 1002 yen. Three packages of tofu cost 8 yen more thantwo cartons of whole milk. What is the cost of a carton ofwhole milk? What is the cost for a package of tofu?

Source: www.tanutech.com/japan/jprice.html (July 2006)

50. Cost of Fast Food Four large cheeseburgers and two choco-late shakes cost a total of $7.90. Two shakes cost 15¢ more thanone cheeseburger. What is the cost of a cheeseburger? What isthe cost of a shake?

51. Computing a Refund The grocery store we use does notmark prices on its goods. My wife went to this store, boughtthree 1-pound packages of bacon and two cartons of eggs, and

paid a total of $7.45. Not knowing that she went to the store,I also went to the same store, purchased two 1-pound pack-ages of bacon and three cartons of eggs, and paid a total of$6.45. Now we want to return two 1-pound packages of baconand two cartons of eggs. How much will be refunded?

52. Blending Coffees A coffee manufacturer wants to market anew blend of coffee that will cost $5 per pound by mixing$3.75-per-pound coffee and $8-per-pound coffee. Whatamounts of the $3.75-per-pound coffee and $8-per-poundcoffee should be blended to obtain the desired mixture? [Hint:Assume the total weight of the desired blend is 100 pounds.]

53. Pharmacy A doctor’s prescription calls for a daily intake ofliquid containing 40 mg of vitamin C and 30 mg of vitamin D.Your pharmacy stocks two liquids that can be used: onecontains 20% vitamin C and 30% vitamin D, the other 40%vitamin C and 20% vitamin D. How many milligrams of eachliquid should be mixed to fill the prescription?

54. Pharmacy A doctor’s prescription calls for the creation ofpills that contain 12 units of vitamin and 12 units of vita-min E. Your pharmacy stocks two powders that can be used to make these pills: one contains 20% vitamin and 30% vitamin E, the other 40% vitamin and 20% vitamin E. Howmany units of each powder should be mixed in each pill?

55. Diet Preparation A 600- to 700-pound yearling horse needs33.0 grams of calcium and 21.0 grams of phosphorus per day for a healthy diet. A farmer provides a combination of rolled oats and molasses to provide those nutrients. Rolled oats provide0.41 grams of calcium per pound and 1.95 grams of phosphorusper pound, while molasses provides 3.35 grams of calcium perpound and 0.36 grams of phosphorus per pound. How manypounds each of rolled oats and molasses should the farmer feedthe yearling in order to meet the daily requirements?

Source: Balancing Rations for Horses, R. D. Setzler, Washing-ton State University

56. Restaurant Management A restaurant manager wants topurchase 200 sets of dishes. One design costs $25 per set, whileanother costs $45 per set. If she only has $7400 to spend, howmany of each design should be ordered?

57. Theater Revenues A Broadway theater has 500 seats, dividedinto orchestra, main, and balcony seating. Orchestra seats sellfor $50, main seats for $35, and balcony seats for $25. If all theseats are sold, the gross revenue to the theater is $17,100. If allthe main and balcony seats are sold, but only half the orchestraseats are sold, the gross revenue is $14,600. How many arethere of each kind of seat?

58. Theater Revenues The Star movie theater charges $8.00 foradults, $4.50 for children, and $6.00 for senior citizens. Oneday the theater sold 405 tickets and collected $2320 in receipts.There were twice as many children’s tickets sold as adult tick-ets. How many adults, children, and senior citizens went to thetheater that day?

59. Prices of Fast Food One group of customers bought 8 deluxehamburgers, 6 orders of large fries, and 6 large colas for $26.10.

B12

B12

B12

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68 Chapter 2 Systems of Linear Equations

Discussion and Writing

67. Make up three systems of two linear equations containing twovariables, one that has no solution, one that has exactly onesolution, and one that has infinitely many solutions. Give thethree systems to a friend to solve and critique.

68. Write a brief paragraph outlining your strategy for solving asystem of two linear equations containing two variables.

A second group ordered 10 deluxe hamburgers, 6 large fries,and 8 large colas and paid $31.60. Is there sufficient informa-tion to determine the price of each food item? If not, constructa table showing the various possibilities. Assume that thehamburgers cost between $1.75 and $2.25, the fries between$0.75 and $1.00. and the colas between $0.60 and $0.90.

60. Prices of Fast Food Use the information given in Problem 59.Suppose that a third group purchased 3 deluxe hamburgers,2 large fries, and 4 large colas for $10.95. Now is there sufficientinformation to determine the price of each food item? If so,determine each price.

61. Supply and Demand Suppose that the quantity supplied Sand quantity demanded D of T-shirts at a concert are given bythe following equations:

where p is the price.

(a) Find the equilibrium price for T-shirts at this concert.

(b) What is the equilibrium quantity?

(c) Graph each equation and label the equilibrium point.

(d) What do you think will eventually happen to the price ofT-shirts if quantity demanded is greater than quantitysupplied?

62. Supply and Demand Suppose that the quantity supplied Sand quantity demanded D of hot dogs at a baseball game aregiven by the following equations:

where p is the price.

(a) Find the equilibrium price for hot dogs at the baseballgame.

(b) What is the equilibrium quantity?

(c) Graph each equation and label the equilibrium point.

(d) What do you think will eventually happen to the price of hotdogs if quantity demanded is less than quantity supplied?

63. IS-LM Model in Economics In economics, the IS curve is alinear equation that represents all combinations of income Yand interest rates r that maintain an equilibrium in the marketfor goods in the economy. The LM curve is a linear equationthat represents all combinations of income Y and interest ratesr that maintain an equilibrium in the market for money in theeconomy. In an economy, suppose the equilibrium level ofincome (in millions of dollars) and interest rates satisfy thesystem of equations

D = 10,000 - 1000p

S = -2000 + 3000p

D = 1000 - 25p

S = -200 + 50p

Find the equilibrium level of income and the interest rate.

64. IS-LM Model in Economics In economics, the IS curve is alinear equation that represents all combinations of income Yand interest rates r that maintain an equilibrium in the marketfor goods in the economy. The LM curve is a linear equationthat represents all combinations of income Y and interestrates r that maintain an equilibrium in the market for moneyin the economy. In an economy, suppose the equilibrium levelof income (in millions of dollars) and interest rates satisfy thesystem of equations

Find the equilibrium level of income and the interest rate.

65. Financial Planning Nathan wants to invest long term the$70,000 profit he made on the sale of his house. Over 10 yearshe hopes to have earnings averaging $5000 per year. Based onthe yearly average return for 10 years ending December 31,2009, his financial advisor recommended three funds: MutualShares at 4%, Franklin Strategic Income at 6%, and FranklinSmall Cap Value at 8% that would allow Nathan to achieve hisgoal while providing diversification.

(a) Prepare a table showing the various ways Nathan canachieve his goal.

(b) What advice would you give him regarding the amount toinvest and the choices available?

Source: Franklin Templeton

66. Financial Planning A married couple wants to invest $60,000long term to supplement their retirement income. They hopeto have earnings averaging $4000 per year. Based on the yearlyaverage return for 10 years ending December 31, 2009, theirfinancial advisor recommended three funds: Franklin TotalReturn at 5%, Franklin Strategic Income at 6%, and FranklinSmall Cap Value at 8% that would allow them to achieve theirgoal while providing diversification.

(a) Prepare a table showing the various ways they can achievetheir goal.

(b) After seeing the table, the couple asked that the amount invested in the Total Return Fund be twice that invested inthe Small Cap Value Fund, how does that change their investment program?

Source: Franklin Templeton

b 0.05Y - 1000r = 10

0.05Y + 800r = 100

b 0.06Y - 5000r = 240

0.06Y + 6000r = 900

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